# 2009 Summer Test 2 solution

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School
Department
Course
Professor 1. (A) “Function f(x) is diﬀerentiable at cif limh0f(c+h)f(c)
hexists. ”
(B) Use your deﬁnition from part (A) to prove f(x) = sin(x)is diﬀerentiable at c.
See the textbook, page 142.
2. (A) Find the equation of the line tangent to the curve 2x3+ 2y3= 9xy at the point
(1,2).
See the textbook, Example 3, page 148.
5(x1) + 2.
(B) Let function g(θ) = |1tan(θ)|have domain θ0,π
2. Calculate the formula
and domain of derivative dg
.
By the chain rule,
dg
=d|u|
du
du
where u= 1 tan(θ)
The derivative of |u|is ±1 and deﬁned whenever u6= 0. Note that,
1tan(θ)>0tan(θ)<1θn0,π
4
1tan(θ)<0tan(θ)>1θπ
4,π
2
Since du
=sec2(θ), this proves,
dg
=
sec2(θ) for θn0,π
4
+ sec2(θ) for θπ
4,π
2,and
undeﬁned for θ=π
4.
(C) Compute limx06x3x
x.
Note that for any x6= 0, 6x3x
x= 3x2x1
x. Since limx03x= 1, it will be suﬃcent to
calculate limx02x1
x(if this limit exists.)
As x0, 2x1
x0
0, and so we may apply L’Hˆopital’s rule:
lim
x0
2x1
x= lim
x0
d
dx (2x1)
d
dx (x)= lim
x0
2xln(2)
1= ln(2).
This proves limx06x3x
x= ln(2).
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